### The Saxl Conjecture for Fourth Powers via the Semigroup Property [article]

Sammy Luo, Mark Sellke
<span title="2016-07-25">2016</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
The tensor square conjecture states that for n ≥ 10, there is an irreducible representation V of the symmetric group S_n such that V ⊗ V contains every irreducible representation of S_n. Our main result is that for large enough n, there exists an irreducible representation V such that V^⊗ 4 contains every irreducible representation. We also show that tensor squares of certain irreducible representations contain (1-o(1))-fraction of irreducible representations with respect to two natural
more &raquo; ... ity distributions. Our main tool is the semigroup property, which allows us to break partitions down into smaller ones.
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